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The product of 0 and anything is $0$, and seems like it would be reasonable to assume that $0 Zero) if $\log 0^0$ worked in your programming language, it's probably because it used the wrong exponentiation convention, and returned $0^0 = 1$. I'm perplexed as to why i have to account for this condition in my factorial function (trying to learn haskell).
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The intention is if you have a number whose magnitude is so small it underflows the exponent, you have no choice but to call the magnitude zero, but you can still salvage the. Doing something wrong implementing an algorithm that explicitly states that $0 \log 0$ is a fib that doesn't mean compute zero times the logarithm of zero, but instead something else (e.g Is a constant raised to the power of infinity indeterminate
Say, for instance, is $0^\\infty$ indeterminate
Or is it only 1 raised to the infinity that is? Inclusion of $0$ in the natural numbers is a definition for them that first occurred in the 19th century The peano axioms for natural numbers take $0$ to be one though, so if you are working with these axioms (and a lot of natural number theory does) then. Why is any number (other than zero) to the power of zero equal to one
Please include in your answer an explanation of why $0^0$ should be undefined. It is possible to interpret such expressions in many ways that can make sense The question is, what properties do we want such an interpretation to have $0^i = 0$ is a good choice, and maybe the only choice that makes concrete sense, since it follows the convention $0^x = 0$
I began by assuming that $\dfrac00$ does equal $1$ and then was eventually able to deduce that, based upon my assumption (which as we know was false) $0=1$
As this is clearly false and if all the steps in my proof were logically valid, the conclusion then is that my only assumption (that $\dfrac00=1$) must be false. In the context of limits, $0/0$ is an indeterminate form (limit could be anything) while $1/0$ is not (limit either doesn't exist or is $\pm\infty$) This is a pretty reasonable way to think about why it is that $0/0$ is indeterminate and $1/0$ is not However, as algebraic expressions, neither is defined
Division requires multiplying by a multiplicative inverse, and. I heartily disagree with your first sentence There's the binomial theorem (which you find too weak), and there's power series and polynomials (see also gadi's answer) For all this, $0^0=1$ is extremely convenient, and i wouldn't know how to do without it
In my lectures, i always tell my students that whatever their teachers said in school about $0^0$.
